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58
Computing Geodesic Paths on Manifolds
 Proc. Natl. Acad. Sci. USA
, 1998
"... The Fast Marching Method [8] is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. A ..."
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Cited by 294 (28 self)
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) steps, where M is the total number of grid points in the domain. The technique hinges on producing numerically consistent approximations to the operators in the Eikonal equation that select the correct viscosity solution; this is done through the use of upwind nite dierence operators. The structure
UPWIND TECHNIQUES AND MIXED FINITE ELEMENTS FOR THE STEADYSTATE BURGERS EQUATION
"... Abstract. In this article the steadystate Burgers equation is studied as a onedimensional simple model for convectiondiusion phenomena. Utilizing nite elements or nite dierences, the numerical solution of the Burgers equation leads to oscillations for small viscosity parameters. In this article ..."
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Abstract. In this article the steadystate Burgers equation is studied as a onedimensional simple model for convectiondiusion phenomena. Utilizing nite elements or nite dierences, the numerical solution of the Burgers equation leads to oscillations for small viscosity parameters. In this article
Optimally accurate secondorder timedomain ¢nite di¡erence scheme for the elastic equation of motion: onedimensional case
"... We previously derived a general criterion for optimally accurate numerical operators for the calculation of synthetic seismograms in the frequency domain (Geller & Takeuchi 1995). We then derived modi¢ed operators for the Direct Solution Method (DSM) (Geller & Ohminato 1994) which satisfy th ..."
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domain ¢nite di¡erence (FD) operators which are second order in space and time using a similar approach. As our FD operators are local, our algorithm is well suited to massively parallel computers. Our approach can be extended to other methods (e.g. pseudospectral) for solving the elastic equation of motion
On Some Upwind Difference Schemes for the Phenomenological SedimentationConsolidation Model
, 2000
"... . In one space dimension, the phenomenological sedimentationconsolidation model reduces to an initialboundary value problem (IBVP) for a nonlinear strongly degenerate convectiondiusion equation with a nonconvex, time dependent ux function. Due to the mixed hyperbolicparabolic nature of the mode ..."
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Cited by 8 (2 self)
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existence and uniqueness results for entropy solutions of IBVPs. The entropy solution framework constitutes the point of departure from which numerical methods can be designed and analysed. The main purpose of this paper is to present and demonstrate several nite dierence schemes which can be used
High Order Finite Di®erencing Schemes and Their Accuracy for CFD
"... This paper investigates di®erent high order ¯nite di®erence schemes and their accuracy for Burgers equation and NavierStokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order di®erence scheme does not necessarily obtain more accurate solutions than a lower ..."
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This paper investigates di®erent high order ¯nite di®erence schemes and their accuracy for Burgers equation and NavierStokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order di®erence scheme does not necessarily obtain more accurate solutions than a
Option Pricing and Linear Complementarity
 Journal of Computational Finance
, 1998
"... Many American option pricing models can be formulated as linear complementarity problems (LCPs) involving partial dierential operators. While recent work with this approach has mainly addressed the model classes where the resulting LCPs are highly structured and can be solved fairly easily, this pap ..."
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Cited by 27 (0 self)
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, this paper discusses a variety of option pricing models that are formulated as partial dierential complementarity problems (PDCPs) of the convectiondiusion kind whose numerical solution depends on a better understanding of LCP methods. Specically, we present secondorder upwind nite dierence schemes
On the method of modi®ed equations. I. Asymptotic analysis of the Euler forward dierence method,
 Departamento de Lenguajes y Ciencias de la Computacion, E.T.S. Ingenieros Industriales, Universidad de Malaga,
, 1997
"... Abstract The method of modi®ed equations is studied as a technique for the analysis of ®nite dierence equations. The nonuniqueness of the modi®ed equation of a dierence method is stressed and three kinds of modi®ed equations are introduced. The ®rst modi®ed or equivalent equation is the natural ps ..."
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Cited by 1 (0 self)
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Abstract The method of modi®ed equations is studied as a technique for the analysis of ®nite dierence equations. The nonuniqueness of the modi®ed equation of a dierence method is stressed and three kinds of modi®ed equations are introduced. The ®rst modi®ed or equivalent equation is the natural
A SIMPLE PROOF FOR THE INVERTIBILITY OF THE LAG POLYNOMIAL OPERATOR
, 2008
"... We provide a proof for the invertibility of the
nite lag polynomial operator in the context of stochastic di¤erence equations, for the case where the polynomial roots lie inside/outside the complex unit circle. We establish invertibility and provide a characterisation for the inverse, using an elem ..."
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We provide a proof for the invertibility of the
nite lag polynomial operator in the context of stochastic di¤erence equations, for the case where the polynomial roots lie inside/outside the complex unit circle. We establish invertibility and provide a characterisation for the inverse, using
Discontinuous Galerkin methods for firstorder hyperbolic problems
"... this paper we consider discontinuous Galerkin (DG) nite element approximations of a model scalar linear hyperbolic equation. We show that in order to ensure continuous stabilization of the method it suces to add a jumppenaltyterm to the discretized equation. In particular, the method does not ..."
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Cited by 30 (9 self)
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for any value of the penalty parameter. As precisely the same jumpterm is used for the purposes of stabilizing DG approximations of advectiondiusion operators, the discretization proposed here can simplify the construction of discontinuous Galerkin nite element approximations of advection
Initial and boundary conditions for the lattice Boltzmann method
 Physical Review E
, 1993
"... A new approach of implementing initial and boundary conditions for the lattice Boltzmann method is presented. The new approach is based on an extended collision operator that uses the gradients of the
uid velocity. The numerical performance of the lattice Boltzmann method is tested on several probl ..."
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Cited by 34 (9 self)
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problems with exact solutions and is also compared to an explicit nite dierence projection method. The discretization error of the lattice Boltzmann method decreases quadratically with ner resolution both in space and in time. The roundo error of the lattice Boltzmann method creates problems unless double
Results 1  10
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58